Lifshitz tails in the 3D Anderson model

Abstract

Consider the 3D Anderson model with a zero mean and bounded i.i.d. random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and $\sigma (E)$ the self energy of the model at energy $E$. For any $\epsilon >0$ and sufficiently small $\lambda$, we derive almost sure localization in the band $E\le -\sigma (0)-{\lambda }^{4-\epsilon }$. In this energy region, we show that the typical correlation length ${\xi }_E$ behaves roughly as $O((|E|-\sigma (E){)}^{-1/2})$, completing the argument outlined in the unpublished work of T. Spencer.

24 pages, 3 figures, to appear in DMJ